Locating saddle points using gradient extremals on manifolds adaptively revealed as point clouds

Area of Science:

• Dynamical Systems and Chaos Theory
• Computational Chemistry
• Differential Geometry

Background:

• Steady states are crucial for understanding dynamical systems.
• High-dimensional systems often simplify to lower-dimensional manifolds due to time scale separation.
• Locating saddle points is essential for analyzing system dynamics and stability.

Purpose of the Study:

• To develop an efficient method for locating saddle points and other fixed points in high-dimensional dynamical systems.
• To address the challenge of analyzing systems evolving on unknown, lower-dimensional manifolds.
• To provide a technique that biases exploration along relevant system trajectories.

Main Methods:

• Utilizes gradient extremals on adaptively sampled point clouds defining unknown Riemannian manifolds.
• Employs manifold learning to discover local coordinates on-the-fly.
• Requires knowledge of a single minimum and sampling capability around arbitrary points.

Main Results:

• Demonstrates effectiveness on the Müller-Brown potential mapped onto a sphere.
• Successfully locates saddle points on an unknown surface.
• Provides a more efficient exploration of the state space compared to exhaustive methods.

Conclusions:

• The proposed method offers an efficient approach to finding saddle points in complex dynamical systems.
• This technique is particularly useful for systems evolving on a priori unknown manifolds.
• The method shows promise for analyzing chemical reaction pathways and other complex phenomena.